Explicit representations of the integral containing the error term in the divisor problem

Abstract

We shall investigate several properties of the integralwith a natural number k, a non-negative integer j and a complex variable θ, where ∆ k (x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the "elementary methods" and the "elementary formulas" to derive convergence properties and explicit representations of this integral with respect to θ for k = 2.

Explicit Representations of the Integral Containing the Error Term in the Divisor Problem Ii

“…We note that the range θ > 5/4 is slightly better than that mentioned in Sitaramachandrarao's paper [7]. However, we stress that the constant '5/4' is the best-possible one that we can obtain by the method used in paper [2], since (x) = (x 1/4 ), and it seems that integral (1.3) would not be convergent absolutely for 1 < θ ≤ 5/4 (cf. Conjecture 1 in [2]).…”

“…However, we stress that the constant '5/4' is the best-possible one that we can obtain by the method used in paper [2], since (x) = (x 1/4 ), and it seems that integral (1.3) would not be convergent absolutely for 1 < θ ≤ 5/4 (cf. Conjecture 1 in [2]). This conjecture means that t −θ (t) log j t would not be Lebesgue-integrable for 1 < θ ≤ 5/4.…”

“…Furthermore, Sitaramachandrarao [7] considered the analytical continuation of this integral by regarding it as a complex function with respect to θ , especially the following formula was derived: 2) which is valid (at least) for θ > 2 − 1/k. In view of the right-hand side in (1.2), we can see that the function…”

“…In [2], we studied the properties of (1.1) for j ≥ 0, especially we treated in detail the case k = 2, that is, the case of the Dirichlet divisor problem. In this case 2 (x) = (x) is written as…”

“…However, our aim of this study, which is the most important point, is that we consider the explicit representation of this function as the 'integral form', not a form of an analytical continuation of this integral. As with the case 1 < θ ≤ 5/4, we studied convergence properties of (1.3) and the explicit representation of this integral in the previous paper [2], and obtained the following theorem.…”